Evolution of Spin-Orbital Entanglement with Increasing Ising Spin-Orbit Coupling - MDPI
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
Article
Evolution of Spin-Orbital Entanglement with
Increasing Ising Spin-Orbit Coupling
Dorota Gotfryd 1,2 , Ekaterina Pärschke 3 , Krzysztof Wohlfeld 1 and Andrzej M. Oleś 2,4, *
1 Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL-02093 Warsaw,
Poland; Dorota.Gotfryd@fuw.edu.pl (D.G.); Krzysztof.Wohlfeld@fuw.edu.pl (K.W.)
2 Institute of Theoretical Physics, Jagiellonian University, Profesora Stanisława Łojasiewicza 11,
PL-30348 Krakow, Poland
3 Institute of Science and Technology Austria, Am Campus 1, A-3400 Klosterneuburg, Austria;
Ekaterina.Plotnikova@phystech.edu
4 Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
* Correspondence: A.M.Oles@fkf.mpg.de
Received: 17 July 2020; Accepted: 23 August 2020; Published: 26 August 2020
Abstract: Several realistic spin-orbital models for transition metal oxides go beyond the
classical expectations and could be understood only by employing the quantum entanglement.
Experiments on these materials confirm that spin-orbital entanglement has measurable consequences.
Here, we capture the essential features of spin-orbital entanglement in complex quantum matter
utilizing 1D spin-orbital model which accommodates SU(2)⊗SU(2) symmetric Kugel-Khomskii
superexchange as well as the Ising on-site spin-orbit coupling. Building on the results obtained
for full and effective models in the regime of strong spin-orbit coupling, we address the question
whether the entanglement found on superexchange bonds always increases when the Ising spin-orbit
coupling is added. We show that (i) quantum entanglement is amplified by strong spin-orbit coupling
and, surprisingly, (ii) almost classical disentangled states are possible. We complete the latter case by
analyzing how the entanglement existing for intermediate values of spin-orbit coupling can disappear
for higher values of this coupling.
Keywords: spin-orbital entanglement; spin-orbit coupling; product state; Mott insulator;
antiferromagnet; ferromagnet; Kugel-Khomskii model; exact diagonalization; entanglement entropy
1. Introduction
Quantum complex matter is characterized by several intertwined degrees of freedom.
Mott insulators with orbital degeneracy belong to this class of materials and are typically characterized
by entangled spin-orbital states, either only in the excited states or in the ground state as well.
Here, we shall present the spin-orbital entanglement in a broader context, summarizing important
results that have been accumulated over the years. With this background established, we will focus
on the recent study [1] of the evolution of spin-orbital entanglement with increasing Ising spin-orbit
coupling, enriching this story with further details.
By definition, quantum entanglement makes information available in one part of the quantum
state dependable on the result of the measurement on the other part of the state [2]. A particular
kind of quantum entanglement, involving quantum spin and orbital operators in the entire quantum
system [3] rather than entanglement between two spatial regions, may appear in the strongly correlated
regime of oxides with orbital degeneracy. The importance of orbitals, besides spins, in the effective
description of such systems was pointed out long ago by Kugel and Khomskii [4]. After this pioneering
work, it has been realized that orbital operators are quantum in nature, enforcing equal treatment
Condens. Matter 2020, 5, 53; doi:10.3390/condmat5030053 www.mdpi.com/journal/condensedmatter
Condens. Matter 2020, 5, 53 2 of 14
with electron spins and producing joint spin-orbital fluctuations [5], both in the ground [6,7] and in
excited [8,9] states. Quantum fluctuations are further amplified by the spin-orbit coupling [10].
In theory, spin-orbital entanglement has been considered in dimerized phases, spin-orbital models
with S = 1/2 spins, both in perovskites [11] and on triangular lattices [12–14], as well as exactly
solvable models such as SU(2)⊗XY model [15], but is weakened in layered oxides suggested as a
possible realization of superconductivity [16]. Several experimental observations in strongly correlated
quantum matter cannot be understood, however, without spin-orbital entanglement. For instance,
the low energy spectral weight in the optical spectroscopy for LaVO3 can be explained only when full
quantum coupling between spins and orbitals is included [17,18], in contrast to LaMnO3 where orbitals
disentangle from large S = 2 spins [19]. Entangled states are also found in disordered spin-orbital
liquid states [20,21] and in Kitaev quantum liquids [22–25], where spatial long-range entanglement [26]
is induced due to strong on-site spin-orbit coupling.
Entanglement in the vanadium perovskites has more manifestations. One of them are the magnon
excitations in the C-type AF (C-AF) phase of YVO3 , where the gap opens along the Γ-Z direction [27],
indicating the dimerization on the bonds along the c-axis which occurs jointly both in spin and orbital
channel at finite temperature [28]. Another experimental confirmation of spin-orbital entanglement
is the evolution of the Néel temperature in the C-AF phase with decreasing ionic radius of R ions in
RVO3 , which is induced by the coupling to the orbital state [29]. Experimentally, the entanglement
is observed in the critical competition between the two spin-orbital ordered states [30] and was
investigated recently in Y0.70 La0.30 VO3 single crystals [31]. Finally, entangled states play a role in
the excited states [32] and may help to identify quantum phase transitions in spin-orbital models [8],
including the transitions at finite magnetic field [33].
Entangled states also play a role in doped spin-orbital systems. One example is colossal
magnetoresistance in manganites, where the orbital polarons modify the ground state in a
one-dimensional (1D) model [34]. Another example is vanadium perovskites with charged defects
which influence occupied orbitals and lead to the collapse of orbital order [35]. The spin-orbital
entangled states were also found among the surface states of topological insulators [36]. More examples
of spin-orbital entanglement could be found in review articles [37,38].
As a matter of fact, both superexchange (i.e., intersite) [3] and spin-orbit coupling (on-site) terms
can promote spin-orbital entanglement, where it may lead to an effective description in terms of the
spin-orbital pseudospins with exotic interactions. The best example is the onset of the Kitaev-like
physics in the iridium oxides [10] in the so-called relativistic Mott insulators [39]. Possible competition
between these two mechanisms (superexchange bonds and spin-orbit) or enhancement of entanglement
by their joint action is a challenging problem. It adds another aspect to the complexity of quantum
matter in high-Tc superconductors [40–44], intimately connected with quantum magnetism in strongly
correlated transition metal oxides [45]. Here, we analyze the spin-orbital entanglement evolution
on a minimal available model, consisting of SU(2)⊗SU(2) symmetric superexchange term [46] and
Ising on-site spin-orbit coupling in one dimension. Below, we recall the recently published thorough
numerical and analytical study [1], supplementing it with the intermediate global stages and previously
not considered cuts, thus completing the story about the disappearance of spin-orbital entanglement
for certain parameter regime.
Condens. Matter 2020, 5, 53 3 of 14
2. The Model and Basic Questions
The full 1D spin-orbital Hamiltonian considered here has the form [1],
H = HSE + HSOC . (1)
The superexchange HSE is a 1D model with two spin (S = 1/2) and two orbital (pseudospin)
(T = 1/2) degrees of freedom per site in a Mott-insulating regime described by SU(2)⊗SU(2) symmetric
expression [47],
HSE = J ∑ {(Si · Si+1 + α) (Ti · Ti+1 + β) − αβ} , (2)
i
where J ≡ 1 is the energy unit and the last term αβ cancels a constant. Spin operators in the scalar
y y
product are, {Si } ≡ {Six , Si , Siz }, and similar for pseudospins, {Ti } ≡ { Tix , Ti , Tiz }. The superexchange
depends on two parameters, α and β, and the spin/pseudospin exchange could be of either sign.
Hence, spin [pseudospin] order could also be of either sign, ferromagnetic (FM), antiferromagnetic
(AF) [ferro-orbital (FO), or alternating-orbital (AO)]. The competition between different channels of
exchange gives a rather rich phase diagram of Equation (2) [47], with three product phases, FM⊗AO,
FM⊗FO, and AF⊗FO, see Figure 1. However, the entanglement plays a crucial role and the phase
diagram includes the fourth spin-orbitally entangled phase near the exactly solvable SU(4) point [47],
where the entanglement entropy of the von Neumann type is finite [48]. In addition, finite (but
much smaller) spin-orbital entanglement characterizes the parameter range where α and β are large
positive, promoting alternating arrangement of spins (pseudospins) with accompanying quantum
fluctuations [48].
1.5
1.0
dimers
AF⊗FO
0.5
β
SU(4)
0.0
-0.5 FM⊗AO
FM⊗FO
-1.0
-1.5 -0.5 0.0 0.5 1.0 1.5
α
Figure 1. Schematic phase diagram of the spin-orbital model (2) in the {α, β} plane inspired by
figure 1 in [48] and figure 7 in [1]. The model has four distinct phases: product disentangled phases
FM⊗AO, AF⊗FO, FM⊗FO, and the phase around the SU(4) point with finite spin-orbital entanglement
on superexchange bonds.
The above model (2) can be derived from the microscopic degenerate Hubbard model with the
diagonal-only hopping t and the Coulomb repulsion U—then, J = 4t2 /U and α = β = 1/4 [49].
The highly symmetric form of spin and pseudospin interactions follows from the degeneracy of
singlet and triplet excited states. These conditions are not fulfilled in widely studied geometries with
corner-sharing MO6 octahedra (M is a transition metal atom) as in perovskites, but the model is a
realistic Kugel–Khomskii model for 1D chains of face-sharing MO6 octahedra [49]. Such materials
Condens. Matter 2020, 5, 53 4 of 14
include for example hexagonal crystals such as BaCoO3 , BaVS3 , or CsCuCl3 . Recently, it was shown that
the specific heat and the susceptibility show anomalies at finite temperature which can be attributed to
phase transitions even in the regime of spin-orbital liquid [50]. Here, we treat it as a generic model
which stands for spin-orbital superexchange and entanglement.
As the competing interaction in Equation (1), we take Ising on-site spin-orbit coupling [1],
HSOC = 2λ ∑ Siz Tiz . (3)
i
It has Z2 -symmetry and favors the opposite orientation of spin hSiz i and orbital h Tiz i moments
at site i. This reduced type of spin-orbit coupling occurs, for instance, when orbital order concerns
two t2g orbitals and the third one is inactive as in LaVO3 or YVO3 [51]. Then, for an idealized case
y
of untilted oxygen octahedra the spin-orbit coupling takes the form 2λSix Ti . Surprisingly, a similar
y
2λSiz Ti form can also be derived for KO2 , where a peculiar form of superexchange includes p orbitals
of molecular oxygen [52]. Taking the symmetry of Hamiltonian (1), these forms are equivalent since
they would modify the quantum states in the same way as the formula in Equation (3), up to the
rotation in the spin/pseudospin sectors. However, in the case of finite orbital-lattice interactions, this
should be carefully studied.
For completeness, we add that, when the full spin-orbit coupling is considered [10], effective
entangled pseudospin J̃ operators replace the sum of spin S and pseudospin T—the J̃ multiplets
then define the ground state and excited states of the system. For instance, spin-orbital entanglement
gives the breakdown of singlets and triplets at the Fermi surface in Sr2 RuO4 [53–55] and their linear
combinations decide about the orbital character of the pairing components [56]. We believe that future
work will also address the role played by entanglement for the d4 ions in more detail where the
pseudospin could vanish and the magnetic moments are hidden in the ground state but would become
active in excited states [57–61].
3. Methods
3.1. Entanglement Entropy and Correlation Functions
In Ref. [1], we have verified that, for (perturbed) product phases of the SU(2)⊗SU(2) model
(see Figure 1), the spin-orbital entanglement evolution with increasing λ experiences only marginal
finite-size effects. Here, we want to show the qualitative effects caused by large spin-orbit coupling
and for that the presented studies of small systems suffice. Therefore, we restrict ourselves to the exact
diagonalization (ED) of a 4-site chain (L = 4) with periodic boundary conditions (PBC). We examine
the spin-orbital entanglement via entanglement entropy of the von Neumann type per site,
1
SvN = − TrS {ρS ln ρS }, (4)
L
where
ρS = TrT |GSihGS| (5)
is the reduced density matrix with the orbital degrees of freedom traced out, estimating the
entanglement by verifying how mixed the reduced state is. Independently, we quantify spin-orbital
entanglement via correlation on a bond hi, i + 1i between nearest neighbors:
L L
1 1
CSO =
L ∑ h(Si · Si+1 )(Ti · Ti+1 )i − L ∑ h Si · Si + 1 i h Ti · Ti + 1 i . (6)
i =1 i =1Condens. Matter 2020, 5, 53 5 of 14
At finite λ, we monitor the response of the system to the Ising spin-orbit coupling via suitable
on-site correlation,
L
1
OSO =
L ∑ hSiz Tiz i . (7)
i =1
Finally, the spin and orbital states are investigated via spin-only and orbital-only correlation,
L
1
S=
L ∑ h S i · S i +1 i , (8)
i =1
L
1
T =
L ∑ h T i · T i +1 i . (9)
i =1
They allow for identifying the product phases, FM⊗AO, FM⊗FO, and AF⊗FO in a
straightforward way.
The entanglement entropy was investigated previously in the SU(2)⊗SU(2) model at λ = 0,
and was found to be finite beyond the product phases where at least one (spin or orbital) component
has suppressed quantum fluctuations in a ferro-state [48]. We have investigated the level of spin-orbital
entanglement in the ground state in the presence of finite spin-orbit coupling and identified small and
large λ regimes along the special α + β = 0 line in the model (1) [1]. While small spin-orbit terms ∝ λ
can be treated in the perturbation theory, below we focus on the more interesting regime of large λ.
3.2. Effective Hamiltonian in the Large λ Limit
In Ref. [1], following the general scheme set in Ref. [10], we have described the low-energy physics
in the λ → ∞ limit by the model utilizing effective pseudospins {J̃i }. Namely, the Hamiltonian (2)
projected onto the low-energy part of HSOC eigenbasis takes the XXZ form:
1 n o
J ∑ J̃i J̃i+1 + J̃i J̃i+1 + 2(α + β) J̃i J̃i+1 .
x x y y z z
Heff = (10)
2 i
z z
It is remarkable that the Ising part of the above Hamiltonian, 2(α + β) J̃i J̃i+1 , vanishes and changes
sign at α + β = 0. As a result, one finds the XY model at β + α = 0, AF Heisenberg model at α + β = 1/2,
and hidden FM Heisenberg at the line α + β = −1/2, denoted by the white dashed lines on Figure 2c.
z z
Below the latter line, one finds a FM Ising ground state, with hJ̃i J̃i+1 i = 1/4 and vanishing quantum
x x y y
fluctuations, hJ̃i J̃i+1 i = hJ̃i J̃i+1 i = 0. For this state, one finds that the mean field approximation for
the spin-spin correlations is exact and CSO = 0 (6), i.e., spin-orbital entanglement vanishes.
In Figure 2c, the highly entangled state exists in a close neighborhood of the β + α = 0 line.
More precisely, highly entangled 1D spin-liquid class of states, also called 1D quantum antiferromagnet
in the literature, could be found in a “quantum stripe” with one sharp boundary at α + β = −1/2 and
system size-dependent soft boundary seemingly approaching AF Heisenberg line α + β = 1/2 for larger
chains, thus roughly fitting within the two outer white dashed lines. We observe that the entanglement
is maximal between the α + β = −1/2 and α + β = 0 lines, and decays at a rate depending on the
system size. This behavior is verified for finite systems, i.e., for rings of L = 4 sites (shown here) and
of L = 8, 12 sites (studied before in [1]). From α + β = 1/2 upwards, we find interpolation between
the 1D Heisenberg antiferromagnet and the classical Néel state in the limit of α + β → ∞ where the
quantum fluctuations are quenched.Condens. Matter 2020, 5, 53 6 of 14
(a) (b) (c) S vN
4.0 0.4
2.0 0.3
0.0 0.2
β
-2.0
0.1
-4.0 λ/J = 0 λ/J = 0. 1 λ/J = 10 6
0.0
-4.0 -2.0 0.0 2.0 4.0 -4.0 -2.0 0.0 2.0 4.0 -4.0 -2.0 0.0 2.0 4.0
α α α
(d) (e) (f) S vN
4.0 0.4
2.0 0.3
0.0 0.2
β
-2.0
0.1
-4.0 λ/J = 0. 2 λ/J = 0. 5 λ/J = 1
0.0
-4.0 -2.0 0.0 2.0 4.0 -4.0 -2.0 0.0 2.0 4.0 -4.0 -2.0 0.0 2.0 4.0
α α α
(g) (h) (i) S vN
4.0 0.4
2.0 0.3
0.0 0.2
β
-2.0
0.1
-4.0 λ/J = 2 λ/J = 5 λ/J = 10
0.0
-4.0 -2.0 0.0 2.0 4.0 -4.0 -2.0 0.0 2.0 4.0 -4.0 -2.0 0.0 2.0 4.0
α α α
Figure 2. Evolution of the entanglement entropy SvN (4) with increasing λ for the Hamiltonian (1)
calculated for a ring of L = 4 sites. Panels present slices of the {α, β} parameter space for a fixed
value of λ/J. (a–c) reproduce surprisingly well the result presented in figure 1 of Ref. [1] for L = 12.
(d–i) reveal intermediate stages between the (b,c). The three white dashed lines represent (from left
to right) hidden FM Heisenberg, XY, and AF Heisenberg models emerging in the λ → ∞ limit at
α + β = −1/2, α + β = 0, and α + β = 1/2 (white dashed lines from bottom to top). For longer chains,
e.g., L = 12, the “quantum stripe” (red region near the α + β = 0 line) in (c) fits between hidden FM
Heisenberg and AF Heisenberg lines. Red dashed lines in the panels indicate the cross section along
which the {α, λ} maps are calculated in Figures 3 and 4.Condens. Matter 2020, 5, 53 7 of 14
4. Results and Discussion
4.1. Gradual Evolution of Spin-Orbital Entanglement with Increasing Ising Spin-Orbit Coupling
Here, we first complement the analysis of Ref. [1] with the data for the entanglement entropy in the
intermediate regime of λ/J, building a global picture of spin-orbital entanglement under increasing
λ. Namely, we add six entanglement entropy scans in Figure 2d–i, fitting in between (b) and (c).
Results presented here for L = 4 include small finite size effects and thus one finds finite spin-orbital
entanglement above the stripe for large λ. Nevertheless, the evolution of perturbed product phases
is qualitatively the same as before for all 4n chains [1], showing that spin-orbital entanglement is
pretty local.
Finite spin-orbital entanglement in the central region of λ = 0 slice (Figure 2a) increases and
spreads out with growing λ, see (b) and (d)–(g). In particular, infinitesimal values of entanglement
entropy appear on the boundaries of perturbed product phases containing either AF or AO component,
see (b,d), and clearly visible on panels (e)–(f). At the same time, FM⊗FO phase does not show
any entanglement, which cannot be induced in this classical state by also classical spin-orbit Ising
coupling (3). Interestingly, the border between completely classical region, originally hosting FM⊗FO
ground state, and the spin-orbitally entangled region gradually changes its shape. The change starts
for infinitesimal λ > 0 but can be resolved starting from panel (d). A sharp boundary, at first restricting
the classical state to the third quarter of (b), moves so that a much wider classical region is visible in (i).
This result is quite close to the classical triangle seen in (c).
There is a huge difference in size between the maximal and minimal spin-orbital entanglement
for varying λ, displayed in Figure 2a–c, correspondingly. In the λ = 0 case, the spin-orbital
term, (Si Si+1 )(Ti Ti+1 ), can produce substantial entanglement in the ground state only in a small
approximately triangular region, where β and α scales of purely spin or purely orbital interactions are
small enough. This corresponds to the critical spin-orbitally entangled region in the phase diagram of
the SU(2)⊗SU(2) model [47]. In the λ → ∞ limit, the “quantum stripe” characterized by the highly
entangled state can be also related to the (Si Si+1 )(Ti Ti+1 ) joint interaction. Namely, when projecting
the Hamiltonian (2) on the lower part of HSOC eigenbasis term by term, one observes that
(Si Si+1 )(Ti Ti+1 ) reduces to the XY term, independent of α and β, while β Si Si+1 and α Ti Ti+1
z z z z
transform into ∝ β J̃i J̃i+1 and ∝ α J̃i J̃i+1 , respectively. As a result, on the α + β = 0 line, the Ising
terms cancel out, leaving the XY model with entangled ground state, and on the entire “quantum
stripe” the ratio Ising/XY supports the highly entangled state. The left boundary of the stripe is well
defined by the α + β = −1/2 line, where FM Heisenberg ground state appears. The degenerate energy
splits immediately below the above line, leaving Ising 2-fold degeneracy.
4.2. Entanglement Evolution for the Quantum Phases along the α + β = 0 Line
Next, we recall the evolution of spin-orbital entanglement on the α + β = 0 line, marked by
the middle diagonal white dashed line in different panels of Figure 2 and investigated originally in
Ref. [1] (see Figure 3 there). This line goes through the two product phases, FM⊗AO and AF⊗FO,
and through the entangled phase which separates them (similar to the white line in Figure 2 at
α + β = 1/2). Two distinct types of evolution emerged on the vertical lines in Figure 3 (under increasing
λ): (i) evolution of type A starting from AF⊗FO or FM⊗AO state with no spin-orbital entanglement
at λ = 0, and (ii) evolution of type B starting from the spin-orbitally entangled critical SU(4) singlet
ground state [48]. Altogether, the behavior of the entanglement entropy and the correlation functions
displayed in Figure 3 for the L = 4 system is very similar to that shown in figure 2 of Ref. [1] for
the L = 12 system, which confirms that, especially for the evolution of type A, the spin-orbital
entanglement is a local phenomenon and the finite size effects are small in the quantities analyzed here.
In case (i), the ground state has large degeneracy at λ = 0 due to FM or FO sector. From the
energetic point of view, the level crossing is at λ = 0 in the degenerate state. Then, λ 6= 0 lifts the
degeneracy and leads to non-degenerate perturbed product states. From then on, all changes in theCondens. Matter 2020, 5, 53 8 of 14
ground state have a crossover character which is visible in the smooth increase of the entanglement
entropy SvN along the line A, see Figure 3d. Finite entanglement is then transferred to the product
phases at increasing λ. However, the most important is that even when the energy scales fully
separate at λ/J 1, finite spin-orbital entanglement is found on superexchange bonds, see Figure 3b.
The finite value of CSO (7) is induced by the (Si · Si+1 + α) (Ti · Ti+1 + β) term to be reduced to the XY
model and freed from α and β-dependent Ising contributions in λ → ∞ limit. Figure 3 shows that this
limit is approximately valid as soon as the non-zero entropy and CSO values spread on α + β = 0 line.
This can be also verified by the maximal response in on-site correlations OSO (7), see Figure 3c.
(a) A B A S vN (b) C SO (c) O SO
10 1 0.00 0.00
0.4
10 0 0.05 0.05
0.3
10 -1 0.10 0.10
λ/J
0.2 0.15 0.15
10 -2
0.1 0.20 0.20
10 -3
0.0 0.25 0.25
1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0
α α α
(d) A (e) B (f) S
10 1 0.50
0.4
10 0 0.25
0.3
10 -1 0.00
λ/J
S vN
0.2
10 -2
0.1 0.25
10 -3
0.0 0.50
10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 -3 10 -2 10 -1 10 0 10 1 10 2 1.0 0.5 0.0 0.5 1.0
λ/J λ/J α
Figure 3. Evolution of the von Neumann spin-orbital entanglement entropy and the three spin-orbital
correlation functions in the ground state of Hamiltonian (1) with α + β = 0 for α ∈ [−1.0, 1.0],
calculated with ED for the ring of L = 4 sites for logarithmically increasing spin-orbit coupling
λ: (a) the von Neumann spin-orbital entanglement entropy SvN (4); (b) the intersite spin-orbital
correlation function CSO (6); (c) the on-site spin-orbit correlation function OSO (7); (f) the spin correlation
function S (8). (d,e) show the von Neumann spin-orbital entanglement entropy SvN (4) obtained for
increasing λ at |α| = 1/2 [cut A in ((a)] and fitted with a logistic function (black thin line), and at α = 0
[cut B in (a)], respectively.
Entanglement entropy SvN in case (ii) starts from a finite value at λ → 0 in spin-orbitally
entangled critical singlet phase [48]. Such singlet correlations are robust along the path B and thus
the spin-orbital entanglement (described by the entropy and the nearest neighbor correlations CSO )
can change only in a discontinuous way, by the kink shown in Figure 3e. We have seen before
that the evolution of spin-orbital entanglement for 4n–site chains occurs along the line of type B
through a series of n kinks [1]. Entanglement gradually gets more robust up to λcrit /J ' 0.2,
above which the highly-entangled state is reached. With this, we completed a short summary of
the main results of Ref. [1].Condens. Matter 2020, 5, 53 9 of 14
4.3. Entanglement Evolution for the Quantum Phases along the α + β = −1 Line
Figure 2 revealed the gradual evolution of the spin-orbital entanglement in λ. Moreover, we
have seen how classical region in both spin and orbital sectors, initially restricted to FM⊗AO phase in
the third quarter of the diagram, gradually spreads out, reaching extended triangular shape within
the boundaries shown in Figure 2c. Within this triangular region, the evolution of FM⊗FO phase
with increasing λ is rather trivial as long as we have the Ising form of the spin-orbit coupling (3).
However, it is not obvious how the lower parts of AF⊗FO or FM⊗AO phases, see Figure 1, approach
the Ising FM ground state below the α + β = −1/2 line (the lowest white dashed line in Figure 2).
Here, we concentrate on the evolution in λ along the representative α + β = −1 line (red dashed
line in Figure 2), including both of the above cases, completing the review of the λ evolution for the
Hamiltonian Equation (1).
We have selected this line because it crosses all three product phases AF⊗FO, FM⊗FO, and
FM⊗AO at λ = 0 (Figure 1) and lies below the “quantum stripe” for large λ. The line runs close enough
to the center of the {α, β} maps to bear quite insightful information. The entanglement evolution in λ
along the line is shown in Figure 4. Panels (a) and (b) show, respectively, the entanglement entropy
and the CSO maps. One can observe that AF⊗FO and FM⊗AO, represented again by cross section C,
at first gain gradually spin-orbital entanglement with growing λ, just like in the evolution along the
line A in Figure 4. S and T correlations confirm that those are indeed perturbed AF⊗FO and FM⊗AO
phases. (S correlation is shown in Figure 4f). However, for some particular value of λ, the growth
of entanglement is sharply cut off. For parallel lines lying deeper within FM⊗FO phase, the effects
discussed here will appear as well but for higher values of λ (see the trends in Figure 2).
The origin of this evolution can be resolved by inspecting the energy spectrum of the L = 4-site
ring, plotted in Figure 4d at α = −2.5 and β = 1.5 and varying λ. Here, we note that for λ > 0 the
ground state degeneracy is completely lifted. It is quite difficult to spot the level crossing suggested by
sharp changes in entanglement. Therefore, in Figure 4g, we analyzed the energy differences between
the excited energies and the ground state energy, ∆E = Ei − E0 , plotted in gray. For convenience,
the ground state energy is renormalized as ∆E = Ei − E0 = 0 and plotted in blue. One observes that
at λ ≈ 8.175 ± 0.025, 2-fold degenerate excited level crosses with the non-degenerate ground state,
leaving 2-fold degenerate ground state. In this way, the perturbed product state becomes an excited
state and gradually gains entanglement, even crossing another excited 2-fold degenerate state for
further increasing λ, while the Ising ferromagnet becomes the ground state.
For completeness, we describe also the evolution of the FM⊗FO state, choosing cross section B at
α = β = −1/2, see Figure 4h. The degenerate 25-fold FM⊗FO level splits immediately when λ > 0,
leaving only 2-fold degeneracy. Consistently, (a) and (b) confirm no spin-orbital entanglement, while
(c) and (h) show that the response of the system to spin-orbit coupling is instantaneous.Condens. Matter 2020, 5, 53 10 of 14
(a) C D C S vN (b) C SO (c) O SO
0.00 0.00
10 1 0.4
0.05 0.05
10 0 0.3
0.10 0.10
10 -1
λ/J
0.2 0.15 0.15
10 -2
0.1 0.20 0.20
10 -3
0.0 0.25 0.25
3 2 1 0 1 2 3 2 1 0 1 2 3 2 1 0 1 2
α α α
(d) C (e) D (f) S
15.0 10.0 0.50
10.0 10 1
5.0 5.0 0.25
0.0 10 0
-5.0 0.0 10 -1 0.00
λ/J
E
1
-10.0 2
10 -2
-15.0 -5.0 0.25
-20.0 2→
10 -3
-25.0 -10.0 0.50
10 -3 10 -2 10 -1 10 0 10 1 10 -3 10 -2 10 -1 10 0 10 1 3 2 1 0 1 2
λ/J λ/J α
(g) C (h) D (i)
2.5
0.1 0.4
2.0
0.08 2 1.5 0.3
0.06 1 A→
1.0 ←C
∆E
S vN
0.04 2
2 0.2
0.5
0.02 2 0.1
1 3 2 0.0
0.0 25 2
-0.5 0.0
7.5 8.0 8.5 9.0 0.0 0.5 1.0 1.5 2.0 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3
λ/J λ/J λ/J
Figure 4. Absence of large spin-orbital entanglement along the red α + β = −1 line in Figure 2 for the
1D ring of L = 4 sites. Top panels show the maps for the (α, λ)-plane: (a) entanglement entropy SvN (4);
(b) spin-orbital CSO correlation (6), (c) on-site OSO correlation (7), and (f) spin-only correlation S (8).
(d,e) present full energy spectrum along cross section C (D), with the ground state energy marked
in blue; (g) the energy differences ∆E = Ei − E0 between the first four excited states and the ground
state around the sharp transition from non-degenerate perturbed AF⊗FO (FM⊗AO) level into 2-fold
degenerate Ising FM ground state; (h) presents the splitting of 25-fold degenerate FM⊗FO ground state
at infinitesimal λ > 0—it gives the 2-fold degenerate Ising FM ground state. In (i), we contrast the
entanglement entropy SvN (4) along line C and line A in Figure 3.
5. Conclusions
In summary, both in Ref. [1] and here, we have accumulated evidence for two distinct regimes
of Ising spin-orbit coupling: (i) when it is weak, i.e., λ < λcrit , one is in the perturbative regime and
the superexchange is almost undisturbed, while (ii) strong spin-orbit coupling λ > λcrit reduces
entanglement due to superexchange but may induce it in the product phases where quantum
fluctuations are finite at λ = 0, in either spin or orbital channel. The fluctuations enhance strongly the
entanglement of already spin-orbital entangled central region. Note that here we define λcrit differently
for the evolution of type A and B in Figure 3 and type C and D in Figure 4.Condens. Matter 2020, 5, 53 11 of 14
For the evolution of type A on the α + β = 0 line, λcrit stands for the dynamical region of rapid
entanglement growth (in logarithmic scale), see Figure 3. It connects the perturbed FM⊗AO and
AF⊗FO phases to the highly entangled state at λ > λcrit via crossover [1]. The evolution of type C on
the α + β = −1 line is more subtle. The perturbed FM⊗AO (AF⊗FO) rapidly gains entanglement in a
smooth way, but the crossover process is interrupted by a level crossing with the Ising ferromagnet.
These transitions seem to be immune to finite-size effects.
For the evolution of type B on the α + β = 0 line, analyzing the spin-orbital entangled critical SU(4)
singlet state through n kinks for L = 4n-site system, we can set λcrit ( L → ∞) ≈ 0.2J [1]. In other words,
finite-size scaling analysis in Ref. [1] suggests that the entanglement growth from the singlet state will
be continuous in the thermodynamic limit, with a phase transition at λcrit ≈ 0.2J. Formally λcrit could
be set to 0, since the large FM (FO) ground state degeneracy at λ = 0 is immediately reduced to 2-fold
degeneracy at λ = 0, matching the Ising state. However, the simplest possible state representing the
ground state energy is disentangled in the whole λ range. The case of FM⊗FO is rather special as the
quantum fluctuations are suppressed in both spin and orbital channel at λ = 0. They are also absent in
the Ising spin-orbit coupling (3), so irrespectively of the actual value of λ, there is no driving force to
generate entangled states. As a result, this phase remains disentangled and has just maximal on-site
spin-orbital correlations OSO = −1/4 which are unable to induce any intersite entanglement.
We emphasize that this will change when spin-orbit coupling is quantum and many-body
states fully develop. For instance, such states are found in two-dimensional (2D) iridates Ba2 IrO4
and Sr2 IrO4 , where it was demonstrated that electron- and hole-doped iridates are fundamentally
different [62]. A similar situation is found in CuAl2 O4 spinel, where spin-orbital entangled Kramers
doublets form and again the electron addition spectra are well described by the Je f f = 1/2 hole
state, whereas electron-removal spectra have a rich multiplet structure [63]. Although the low-energy
physics depends on the iridate geometry, it would be challenging to investigate in a similar way the
interplay of spin-orbital superexchange with quantum spin-orbit coupling.
We also reported an interesting “emergence” of the spin-orbital entanglement along the α + β = −1
line in the Ising limit. In this case, both for the negligibly small λ as well as for the large λ > λcrit the
spin-orbital entanglement vanishes. Surprisingly, however, for a moderate value of λ (and a finite value
of α) the spin-orbital entanglement becomes finite. This shows the intricate complexity of the studied
system. Crucially, the analysis at λ > λcrit is consistent with the phase diagram of the effective XXZ
Hamiltonian (10). In particular, the ground state with vanishing spin-orbital entanglement appears
when the quantum fluctuations vanish in the ground state of an effective model. This is the case of an
Ising ferromagnet obtained at α + β ≤ −1/2.
Finally, we argue that the results presented here may play an important role (after extension) in
the understanding of the strongly correlated systems with non-negligible spin-orbit coupling—such as,
e.g., the 5d iridates, 4d ruthenates, 3d vanadates, the 2p alkali hyperoxides, and other to-be-synthesized
materials. More discussion of the entanglement in these materials was given before [1]. As discussed
also in Ref. [1], the results presented for the 1D model can be (on a qualitative level) extended to
the higher-dimensional hypercubic lattice. This is due to the fact that the mapping to the effective
model is valid independently of the dimension. Therefore, the spin-orbital correlation function (and
consequently the spin-orbital entanglement) is nonzero for the similar range of model parameters in
2D or three-dimensional systems as for the 1D model reported here.
Author Contributions: All authors participated equally in the formulation of research ideas; calculations, D.G.;
derivation of the effective strong coupling model, E.P.; results selection and paper writing, D.G. and A.M.O.;
graphical visualization, D.G.; ultimate formulation—review, citations, and correspondence, A.M.O. and K.W.
All authors have read and agreed to the published version of the manuscript.
Funding: This research was supported by Narodowe Centrum Nauki (NCN, Poland) under Projects No.
2016/23/B/ST3/00839 and No. 2016/22/E/ST3/00560. E.M.P. was supported by the European Union’s Horizon
2020 research and innovation program under the Maria Skłodowska-Curie Grant No. 754411.Condens. Matter 2020, 5, 53 12 of 14
Acknowledgments: It is our great pleasure to thank Wojtek Brzezicki, Jiří Chaloupka, Daniel I. Khomskii, and
Klim Kugel for many insightful discussions. A.M.O. is grateful for the Alexander von Humboldt Foundation
Fellowship (Humboldt–Forschungspreis).
Conflicts of Interest: The authors declare no conflict of interest.
References
1. Gotfryd, D.; Pärschke, E.M.; Chaloupka, J.; Oleś, A.M.; Wohlfeld, K. How spin-orbital entanglement depends
on the spin-orbit coupling in a Mott insulator. Phys. Rev. Res. 2020, 2, 013353. [CrossRef]
2. Bentsson, I.; Zyczkowski, K. Geometry of Quantum States: An Introduction to Quantum Entanglement;
Cambridge University Press: Cambridge, UK, 2006.
3. Oleś, A.M.; Horsch, P.; Feiner, L.F.; Khaliullin, G. Spin-Orbital Entanglement and Violation of the
Goodenough-Kanamori Rules. Phys. Rev. Lett. 2006, 96, 147205. [CrossRef] [PubMed]
4. Kugel, K.I.; Khomskii, D.I. Jahn-Teller Effect and Magnetism: Transition Metal Compounds. Usp. Fiz. Nauk
1982, 25, 621–664. [CrossRef]
5. Feiner, L.F.; Oleś, A.M.; Zaanen, J. Quantum Melting of Magnetic Order due to Orbital Fluctuations.
Phys. Rev. Lett. 1997, 78, 2799–2802. [CrossRef]
6. Chen, Y.; Wang, Z.D.; Li, Y.Q.; Zhang, F.C. Spin-orbital entanglement and quantum phase transitions in a
spin-orbital chain with SU(2)×SU(2) symmetry. Phys. Rev. B 2007, 75, 195113. [CrossRef]
7. Brzezicki, W.; Dziarmaga, J.; Oleś, A.M. Noncollinear Magnetic Order Stabilized by Entangled Spin-Orbital
Fluctuations. Phys. Rev. Lett. 2012, 109, 237201. [CrossRef]
8. You, W.-L.; Horsch, P.; Oleś, A.M. Entanglement driven Phase Transitions in Spin-Orbital Models. New J. Phys.
2015, 17, 083009. [CrossRef]
9. Snamina, M.; Oleś, A.M. Magnon Dressing by Orbital Excitations in Ferromagnetic Planes of K2 CuF4 and
LaMnO3 . New J. Phys. 2019, 21, 023018. [CrossRef]
10. Jackeli, G.; Khaliullin, G. Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a
Quantum Compass and Kitaev Models. Phys. Rev. Lett. 2009, 102, 017205. [CrossRef]
11. Khaliullin, G.; Maekawa, S. Orbital Liquid in Three-Dimensional Mott Insulator: LaTiO3 . Phys. Rev. Lett.
2000, 85, 3950–3953. [CrossRef]
12. Normand, B.; Oleś, A.M. Frustration and Entanglement in the t2g Spin–Orbital Model on a Triangular Lattice:
Valence–Bond and Generalized Liquid States. Phys. Rev. B 2008, 78, 094427. [CrossRef]
13. Normand, B. Multicolored quantum dimer models, resonating valence-bond states, color visons, and the
triangular-lattice t2g spin-orbital system. Phys. Rev. B 2011, 83, 064413. [CrossRef]
14. Chaloupka, J.; Oleś, A.M. Spin-Orbital Resonating Valence-Bond Liquid on a Triangular Lattice: Evidence
from Finite Cluster Diagonalization. Phys. Rev. B 2011, 83, 094406. [CrossRef]
15. Brzezicki, W.; Dziarmaga, J.; Oleś, A.M. Topological Order in an Entangled SU(2)⊗XY Spin-Orbital Ring.
Phys. Rev. Lett. 2014, 112, 117204. [CrossRef] [PubMed]
16. Chaloupka, J.; Khaliullin, G. Orbital Order and Possible Superconductivity in LaNiO3 /LaMO3 Superlattices.
Phys. Rev. Lett. 2008, 100, 016404. [CrossRef]
17. Miyasaka, S.; Okimoto, Y.; Tokura, Y. Anisotropy of Mott-Hubbard Gap Transitions due to Spin and Orbital
Ordering in LaVO3 and YVO3 . J. Phys. Soc. Jpn. 2002, 71, 2086–2089. [CrossRef]
18. Khaliullin, G.; Horsch, P.; Oleś, A.M. Theory of Optical Spectral Weights in Mott Insulators with Orbital
Degrees of Freedom. Phys. Rev. B 2004, 70, 195103. [CrossRef]
19. Kovaleva, N.N.; Oleś, A.M.; Balbashov, A.M.; Maljuk, A.; Argyriou, D.N.; Khaliullin, G.; Keimer, B.
Low-energy Mott-Hubbard excitations in LaMnO3 probed by optical ellipsometry. Phys. Rev. B
2010, 81, 235130. [CrossRef]
20. Laurita, N.J.; Deisenhofer, J.; Pan, L.-D.; Morris, C.M.; Schmidt, M.; Johnsson, M.; Tsurkan, V.; Loidl, A.;
Armitage, N.P. Singlet-Triplet Excitations and Long-Range Entanglement in the Spin-Orbital Liquid Candidate
FeSc2 S4 . Phys. Rev. Lett. 2015, 114, 207201. [CrossRef]
21. Man, H.Y.; Halim, M.; Sawa, H.; Hagiwara, M.; Wakabayashi, Y.; Nakatsuji, S. Spin-orbital entangled liquid
state in the copper oxide Ba3 CuSb2 O9 . J. Phys. Condens. Matter 2018, 30, 443002. [CrossRef]
22. Becker, M.; Hermanns, M.; Bauer, B.; Garst, M.; Trebst, S. Spin-orbit physics of j = 1/2 Mott insulators on the
triangular lattice. Phys. Rev. B 2015, 91, 155133. [CrossRef]Condens. Matter 2020, 5, 53 13 of 14
23. Hermanns, M.; Kimchi, I.; Knolle, J. Physics of the Kitaev Model: Fractionalization, Dynamic Correlations,
and Material Connections. Ann. Rev. Cond. Mat. Phys. 2018, 9, 17–33. [CrossRef]
24. Rusnačko, J.; Gotfryd, D.; Chaloupka, J. Kitaev-like Honeycomb Magnets: Global Phase Behavior and
Emerging Effective Models. Phys. Rev. B 2019, 99, 064425. [CrossRef]
25. Natori, W.M.H.; Knolle, J. Dynamics of a Two-Dimensional Quantum Spin-Orbital Liquid: Spectroscopic
Signatures of Fermionic Magnons. Phys. Rev. Lett. 2020, 125, 067201. [CrossRef]
26. Takagi, H.; Takayama, T.; Jackeli, G.; Khaliullin, G.; Nagler, S.E. Concept and realization of Kitaev quantum
spin liquids. Nat. Rev. Phys. 2019, 1, 264–280. [CrossRef]
27. Ulrich, C.; Khaliullin, G.; Sirker, J.; Reehuis, M.; Ohl, M.; Miyasaka, S.; Tokura, Y.; Keimer, B. Magnetic Neutron
Scattering Study of YVO3 : Evidence for an Orbital Peierls State. Phys. Rev. Lett. 2003, 91, 257202. [CrossRef]
28. Sirker, J.; Herzog, A.; Oleś, A.M.; Horsch, P. Thermally Activated Peierls Dimerization in Ferromagnetic Spin
Chains. Phys. Rev. Lett. 2008, 101, 157204. [CrossRef]
29. Horsch, P.; Oleś, A.M.; Feiner, L.F.; Khaliullin, G. Evolution of Spin-Orbital-Lattice Coupling in the RVO3
Perovskites. Phys. Rev. Lett. 2008, 100, 167205. [CrossRef]
30. Fujioka, J.; Yasue, T.; Miyasaka, S.; Yamasaki, Y.; Arima, T.; Sagayama, H.; Inami, T.; Ishii, K.; Tokura, Y.
Critical competition between two distinct orbital-spin ordered states in perovskite vanadates. Phys. Rev. B
2010, 82, 144425. [CrossRef]
31. Yan, J.-Q.; Tian, W.; Cao, H.B.; Chi, S.; Ye, F.; Llobet, A.; Puretzky, A.; Chen, Q.; Ma, J.; Ren, Y.; et al. Lattice
distortion in the spin-orbital entangled state in RVO3 perovskites. Phys. Rev. B 2019, 100, 184423. [CrossRef]
32. Chen, C.C.; van Veenendaal, M.; Devereaux, T.P.; Wohlfeld, K. Fractionalization, entanglement, and separation:
Understanding the collective excitations in a spin-orbital chain. Phys. Rev. B 2015, 91, 165102. [CrossRef]
33. Ronquillo, D.C.; Vengal, A.; Trivedi, N. Signatures of magnetic-field-driven quantum phase transitions in the
entanglement entropy and spin dynamics of the Kitaev honeycomb model. Phys. Rev. B 2019, 99, 140413(R).
[CrossRef]
34. Daghofer, M.; Oleś, A.M.; von der Linden, W. Orbital Polarons versus Itinerant e g Electrons in Doped
Manganites. Phys. Rev. B 2004, 70, 184430. [CrossRef]
35. Avella, A.; Oleś, A.M.; Horsch, P. Defect-Induced Orbital Polarization and Collapse of Orbital Order in Doped
Vanadium Perovskites. Phys. Rev. Lett. 2019, 122, 127206. [CrossRef]
36. Yaji, K.; Kuroda, K.; Toyohisa, S.; Harasawa, A.; Ishida, Y.; Watanabe, S.; Chen, C.T.; Kobayashi, K.; Komori, F.;
Shin, S. Spin-dependent quantum interference in photoemission process from spin-orbit coupled states.
Nat. Commun. 2017, 8, 14588. [CrossRef]
37. Oleś, A.M. Fingerprints of Spin-Orbital Entanglement in Transition Metal Oxides. J. Phys. Condens. Matter
2012, 24, 313201. [CrossRef]
38. Brzezicki, W. Spin, orbital and topological order in models of strongly correlated electrons. J. Phys.
Condens Matter 2020, 32, 023001. [CrossRef]
39. Witczak-Krempa, W.; Chen, G.; Kim, Y.B.; Balents, L. Correlated Quantum Phenomena in the Strong Spin-Orbit
Regime. Ann. Rev. Cond. Mat. Phys. 2014, 5, 58–82. [CrossRef]
40. Bianconi, A. Superstripes and complexity in High-Temperature Superconductors. J. Supercond. Nov. Magn.
2014, 27, 909–912. [CrossRef]
41. Keimer, B.; Kivelson, S.A.; Norman, M.R.; Uchida, S.; Zaanen, J. From quantum matter to high-temperature
superconductivity in copper oxides. Nature 2015, 518, 179–186. [CrossRef]
42. Campi, G.; Bianconi, A. High-Temperature Superconductivity in a Hyperbolic Geometry of Complex Matter
from Nanoscale to Mesoscopic Scale. J. Supercond. Nov. Magn. 2016, 29, 627–631. [CrossRef]
43. Bussmann-Holder, A.; Kohler, J.; Simon, A.; Whangbo, M.-H.; Bianconi, A.; Perali, A. The Road Map toward
Room-Temperature Superconductivity: Manipulating Different Pairing Channels in Systems Composed of
Multiple Electronic Components. Condens. Matter 2017, 2, 24. [CrossRef]
44. Bianconi, A. Lifshitz Transitions in Multi-band Hubbard Models for Topological Superconductivity in
Complex Quantum Matter. J. Supercond. Novel Magn. 2018, 31, 603–610. [CrossRef]
45. Khomskii, D.I. Transition Metal Compounds; Cambridge University Press: Cambridge, UK, 2014.
46. Pati, S.K.; Singh, R.R.P.; Khomskii, D.I. Alternating Spin and Orbital Dimerization and Spin-Gap Formation
in Coupled Spin-Orbital Systems. Phys. Rev. Lett. 1998, 81, 5406–5409. [CrossRef]
47. Itoi, C.; Qin, S.; Affleck, I. Phase diagram of a one-dimensional spin-orbital model. Phys. Rev. B 2000, 61, 6747.
[CrossRef]Condens. Matter 2020, 5, 53 14 of 14
48. Lundgren, R.; Chua, V.; Fiete, G.A. Entanglement entropy and spectra of the one-dimensional Kugel–Khomskii
model. Phys. Rev. B 2012, 86, 224422. [CrossRef]
49. Kugel, K.I.; Khomskii, D.I.; Sboychakov, A.O.; Streltsov, S.V. Spin-orbital interaction for face-sharing octahedra:
Realization of a highly symmetric SU(4) model. Phys. Rev. B 2015, 91, 155125. [CrossRef]
50. Valiulin, V.E.; Mikheyenkov, A.V.; Kugel, K.I.; Barabanov, A.F. Thermodynamics of Symmetric Spin-Orbital
Model: One- and Two-Dimensional Cases. JETP Lett. 2019, 109, 546–551. [CrossRef]
51. Horsch, P.; Khaliullin, G.; Oleś, A.M. Dimerization versus Orbital Moment Ordering in a Mott Insulator YVO3 .
Phys. Rev. Lett. 2003, 91, 257203. [CrossRef]
52. Solovyev, V.I. Spin-orbital superexchange physics emerging from interacting oxygen molecules in KO2 .
New J. Phys. 2008, 10, 013035. [CrossRef]
53. Veenstra, C.N.; Zhu, Z.-H.; Raichle, M.; Ludbrook, B.M.; Nicolaou, A.; Slomski, B.; Landolt, G.; Kittaka, S.;
Maeno, Y.; Dil, J.H.; et al. Spin-Orbital Entanglement and the Breakdown of Singlets and Triplets in Sr2 RuO4
Revealed by Spin- and Angle-Resolved Photoemission Spectroscopy. Phys. Rev. Lett. 2014, 112, 127002.
[CrossRef] [PubMed]
54. Bahr, S.; Alfonsov, A.; Jackeli, G.; Khaliullin, G.; Matsumoto, A.; Takayama, T.; Takagi, H.;
Büchner, B.; Kataev, V. Low-energy magnetic excitations in the spin-orbital Mott insulator Sr2 IrO4 .
Phys. Rev. B 2014, 89, 180401(R). [CrossRef]
55. Zhang, G.; Gorelov, E.; Sarvestani, E.; Pavarini, E. Fermi Surface of Sr2 RuO4 : Spin-Orbit and Anisotropic
Coulomb Interaction Effects. Phys. Rev. Lett. 2016, 116, 106402. [CrossRef]
56. Oleś, A.M.; Wohlfeld, K.; Khaliullin, G. Orbital Symmetry and Orbital Excitations in High-Tc Superconductors.
Condens. Matter 2019, 4, 46. [CrossRef]
57. Khaliullin, G. Excitonic Magnetism in Van Vleck-type d4 Mott Insulators. Phys. Rev. Lett. 2013, 111, 197201.
[CrossRef]
58. Souliou, S.-M.; Chaloupka, J.; Khaliullin, G.; Ryu, G.; Jain, A.; Kim, B.J.; Le Tacon, M.; Keimer, B. Raman
scattering from Higgs mode oscillations in the two-dimensional antiferromagnet Ca2 RuO4 . Phys. Rev. Lett.
2017, 119, 067201. [CrossRef] [PubMed]
59. Svoboda, C.; Randeria, M.; Trivedi, N. Effective magnetic interactions in spin-orbit coupled d4 Mott insulators.
Phys. Rev. B 2017, 95, 014409. [CrossRef]
60. Li, F.-Y.; Chan, G. Spin-orbital entanglement in d8 Mott insulators: Possible excitonic magnetism in diamond
lattice antiferromagnets. Phys. Rev. B 2019, 100, 045103. [CrossRef]
61. Chaloupka, J.; Khaliullin, G. Highly frustrated magnetism in relativistic d4 Mott insulators: Bosonic analog of
the Kitaev honeycomb model. Phys. Rev. B 2019, 100, 224413. [CrossRef]
62. Pärschke, E.M.; Wohlfeld, K.; Foyevtsova, K.; van den Brink, J. Correlation induced electron-hole asymmetry
in quasi- two-dimensional iridates. Nat. Commun. 2017, 8, 686. [CrossRef] [PubMed]
63. Kim, C.H.; Baidya, S.; Cho, H.; Gapontsev, V.V.; Streltsov, S.V.; Khomskii, D.I.; Park, J.-G.; Go, A.; Jin, H.
Theoretical evidence of spin-orbital-entangled Jeff=12 state in the 3d transition metal oxide CuAl2 O4 .
Phys. Rev. B 2019, 100, 161104(R). [CrossRef]
© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access
article distributed under the terms and conditions of the Creative Commons Attribution
(CC BY) license (http://creativecommons.org/licenses/by/4.0/).You can also read
